Platonic Solids

7 6 the octahedron 8 faces 12 edges 6 vertices The octahedron is made of eight equilateral triangles, four meeting at every vertex. Plato considered the octahedron an intermediary between the tetrahedron, or Fire, and the icosahedron, or Water and thus ascribed it to the element of Air. The octahedron has six 2fold axes passing through opposite edges, four 3fold axes through its face centres and three 4fold axes through opposite vertices below. Solids combining these rotation axes display octahedral symmetry. Greek writings attribute the discovery of the octahedron and icosahedron to Theaetetus of Athens 417 BC 369 BC. Book XIII of Euclids Elements see page 14 is thought to be based on Theaetetus work on the regular solids. The octahedrons circumradius is bigger than its inradius by a factor of 3 see page 55. The same relationship occurs between the circumradius and inradius of the cube, and between the circumradius and midradius and the midradius and inradius of the tetrahedron. The tetrahedron, the octahedron and the cube are all found in the mineral kingdom. Mineral diamonds and common fluorite crystals often form octahedra.